Estimate angular displacement of a free moving pendulum Given a free moving pendulum I need to calculate to approximate the angular displacement using a Kalman Filter.
For this I have $2$ sensors:


*

*A Radar that can calculate the distance to the weight attached to pendulum and the horizontal speed and

*a gyroscope that can measure the angular velocity of the pendulum.


I was able to understand how a Kalman Filter works. I don't understand in this scenario how the radar will help with the estimation.


*

*Isn't enough to know only the angular velocity, $g$ and the length of the wire to estimate the displacement ?.

*How can I use the measurements of the radar ?.

 A: Once you have the length of the pendulum (are you given it?) the two sensors are redundant.  You can use them to reduce the overall error.  Each one gives you a measure of the angular displacement, which you can then combine.  Usually you weight them with the inverse square of the standard deviation because that gives the best overall estimator if the errors are normal.  
If the sensors give a time history, you can use the period to get the length.
A: Yes you are right. In theory, you are able to calculate angular position using only "the angular velocity, $g$ and the length of the wire"; and radar is redundant. But the key of Kalman filter to fuse/combine (seemingly) redundant information. Let's say you have 2 ways to calculate the angular position. In practice, none of them is truly accurate, hence with noises described by some probability distributions. Therefore, Kalman filter is here to fuse (combine) 2 good estimations to 1 better estimation, if noises are Gaussian $\mathcal{N}(\mu, \sigma^2)$.
To use radar sensor data, if angular position estimation from wire etc is $\hat{\theta}_1$ with variance $\sigma_1^2$, and estimation from radar is $\hat{\theta}_2$ with variance $\sigma_2^2$. Assuming both errors are Gaussian. Then the estimation by Kalman filter is $\hat{\theta}=\hat{\theta}_1 + \frac{\sigma_1^2}{\sigma_1^2 + \sigma_2^2} (\hat{\theta}_2-\hat{\theta}_1)$.
